Abstract
We study notions of absolute continuity for functions defined on $\mathbb{R}^n$ similar to the notion of $\alpha$-absolute continuity in the sense of Bongiorno. We confirm a conjecture of Malý that 1-absolutely continuous functions do not need to be differentiable a.e., and we show several other pathological examples of functions in this class. We establish some containment relations of the class $1\textit{-} AC_{\rm WDN}$ which consits of all functions in $1\textit{-}AC$ which are in the Sobolev space $W^{1,2}_{loc}$, are differentiable a.e. and satisfy the Luzin (N) property, with previously studied classes of absolutely continuous functions.
Citation
Michael Dymond. Beata Randrianantoanina. Huaqiang Xu. "On Interval Based Generalizations of Absolute Continuity for Functions on \(\mathbb{R}^{n}\)." Real Anal. Exchange 42 (1) 49 - 78, 2017.